Question

The area of the region bounded by |x| + |y| = 1, x ≥ 0 y ≥ 0 is :

• A. \(\frac{1}{4}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{3}{2}\)
• D. \(\frac{3}{4}\)

Answer 1

The Correct Option is B

The area of the region bounded by the equation \(|x| + |y| = 1\), given the constraints \(x \geq 0\) and \(y \geq 0\), can be found as follows:

This equation represents a square centered at the origin with vertices at (1,0), (0,1), (-1,0), and (0,-1). We are interested in the area within this square that lies in the first quadrant (i.e., where \(x \geq 0\) and \(y \geq 0\)).

The square is divided into four congruent right triangles by the lines \(x = 0\), \(y = 0\), and the diagonals \(x = -y\) and \(x = y\).

In the first quadrant, the equation simplifies to \(x + y = 1\) with intercepts (1,0) and (0,1), forming a right triangle with these two intercepts on the axes and the hypotenuse along the line.

The base and the height of this right triangle are both 1 unit long.

Therefore, the area of the triangle is:

\(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}\)

Thus, the area of the region bounded by the given constraints is \(\frac{1}{2}\).